On the lower bound front, Gonen and Oshman [1]} showed lower bounds for finding subgraphs that are created from smaller ones using certain allowed operations. Fischer, Gonen, Kuhn, and Oshman [2]} showed that for any \(p\ge 4\) , there exists a subgraph \(H\)  of size \(p\)  such that \(H\)  detection requires \(\tilde{\Omega }(n^{2-\Theta (1/p)})\)  rounds. Eden, Fiat, Fischer, Kuhn, and Oshman [3]} showed that this complexity is roughly the right one, by providing an upper bound of \(\tilde{O}(n^{2-2/(3p+1)}+o(1))\)  rounds. In particular, their result shows that there does not exist a constant-sized subgraph for which detection requires a truly quadratic number of rounds.
